Dimension reduction is one of the most highlighted problems in high-dimensional data analysis and visualization. Algorithms are usually designed to optimize cost functions that formalize certain criteria such as maximizing projected variance for the case of principal component analysis. We start by examining the cost function of Fisher’s linear discriminant analysis (LDA) and reveal that the objective function is closely related to well-known statistics from multivariate analysis of variance (MANOVA). Motivated by the association, we propose a general framework for linear dimension reduction under the supervised regime to find an optimal embedding or projection matrix that carries information from a class of hypothesis testing procedures. As a black-box algorithm for potentially non-differentiable and complex cost functions, simulated annealing (SA) is modified on Stiefel manifold of projection matrices. We also present extension to multi-class scenarios, merging heterogeneous information from distinct procedures, and scalable computation via median of subset projections on Grassmann manifold. Simulation results show that the proposed method is competitive to existing methods.